The precise definition of a latent factor, and therefore the precise definition of the intrinsic dimensionality, can be quite challenging. For example, Principal Component Analysis (PCA) (which we will discuss in detail in the sequel) defines latent factors in terms of linear projections and orthogonality. At the other extreme, one can consider quite complicated scenarios involving fractal dimensions and space filling curves [1]}. In this text we take a middle of the road approach, and focus on non-linear, but smooth, manifolds.
Such an approach is quite popular, and such DR and DE techniques are used in diverse domains such as engineering, astronomy[2]}[3]}, biology[4]}[5]}, remote sensing[6]}[7]}, economics[8]}[9]}, social media[10]}[11]}, and finance[12]}[13]}; and this class of techniques has a large extant literature (see, e.g., [14]}, [1]} and references therein).
